@Doris: I don't own Random House "Webster's" Unabridged (which I believe is actually the same main text as Random House unabridged, 2nd ed., 1987), only its 1966 ancestor, Random House unabridged (1st ed.), which is also very good. However, my Webster's Third unabridged (1961; addenda through 1981) is of similar vintage and also defines only 'sixty-four-dollar question' as an entry.
In any case, both those dictionaries, though classics, seem to be behind the curve on this term. 'Sixty-four-thousand-dollar question,' from the later TV show, did actually prove to be the more popular version, and the only one I'm aware of hearing these days, even from older people (born from 1930s on).
In fact, the most recent large AE dictionary that I know of, NOAD (New Oxford American Dictionary, 2001), lists only the later version:
sixty-four thousand dollar question - (informal) something that is not known and on which a great deal depends. (Origin) 1950s: from a question posed for the top prize in a television quiz show of the same name.
I can't condone their sloppy omission of the other two mandatory hyphens, and even on short acquaintance with NOAD I already have some broader reservations about their editorial judgment. But it seems entirely reasonable of them to acknowledge the modern form of the phrase, and I would suggest that LEO do the same, probably by simply listing both versions.
Do you guys have access to a database such as Nexis that searches only published texts such as newspapers and magazines? I don't, but I would guess that the version with 'thousand' is more frequent there too (and that all three hyphens should appear in the more reputably edited sources).